\subsection{}
In \cite{AndersonBCS}, there is time reversal symmetry, but in three species problem, it becomes more complicated due to spin.  We have the symmetry of simple reverse in space though in hamiltonian.  

\subsection{Estimation of $\av{b^{\dagger}_{\vk}c_{\vk+\vq}}$}
This quantity pops up in equation of motion.  This quantity for the equilibrium is $v_{k}w_{k}$ for q=0, can be estimated.  At low k, where $v_{k}$ is large, but $w_{k}$ is generally small, so it is generally much smaller than 

\subsection{Tony's comment}
He asked me to work out the lowest order correction in mean-field, in fermionic excitation (Bogoliubov approch maybe?), and collective excitation.  That should be enough. 